Denseness of certain smooth Lévy functionals in $\DD_{1,2}$

Details Denseness of certain smooth Lévy functionals in $\DD_{1,2}$ Denseness of certain smooth Lévy functionals in $\DD_{1,2}$

2008-05-30

arxiv.org/abs/0805.4704v1

Mathematics

Probability

16 pages

Scientific paper

The Malliavin derivative for a L\'evy process $(X_t)$ can be defined on the space $\DD_{1,2}$ using a chaos expansion or in the case of a pure jump process also via an increment quotient operator \cite{sole-utzet-vives}. In this paper we define the Malliavin derivative operator $\D$ on the class $\mathcal{S}$ of smooth random variables $f(X_{t_1}, ..., X_{t_n}),$ where $f$ is a smooth function with compact support. We show that the closure of $L_2(\Om) \supseteq \mathcal{S} \stackrel{\D}{\to} L_2(\m\otimes \mass)$ yields to the space $\DD_{1,2}.$ As an application we conclude that Lipschitz functions map from $\DD_{1,2}$ into $\DD_{1,2}.$

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